nLab
Gabriel composition of filters

Given a filterFF of ideals in a ring RR, a left RR-moduleMM is FF-torsion (Gabriel’s terminology: FF-negligible) if for any m∈Mm\in M there exists a LL in FF which annihilates it: Lm=0L m=0.

Given two filters F,GF,G in the latticeIlRI_l R of left ideals on RR, one defines their Gabriel composition (or Gabriel product) F•GF\bullet G as the set of all left RR-ideals LL such that there is KK in GG such that K/LK/L is FF-torsion. Gabriel composition of filters corresponds to the Gabriel multiplication of the corresponding torsion classes considered as strictly full subcategories. A Gabriel composition of uniform filters is uniform.